Within the general theory for optimal control of VI [1]}, [2]},
the main challenge consists in the derivation of optimality conditions.
It can be studied by proper approximation of VI by regularized equations
and taking the limit as the regularization parameter tends to zero.
The corresponding methods for optimal control of obstacle problems
can be found in [3]}, [4]} using augmented Lagrangians,
in e.g. [5]}, [6]} for a Moreau–Yosida regularization, and in [7]}
based on a Lavrentiev regularization, for the latter see [8]}, [9]}.
Furthermore we cite [10]}, [11]} for control of non-smooth and non-convex
functionals, [12]} for boundary control, and [13]}, [14]}
for control of quasi- and hemi-VI.
Shape optimization for free-interface identification with obstacle-type VI
using adjoints was developed recently by [15]}, [16]}.
The common difficulty is a lack of regularity that needs
assumptions on a solution in order to take the limit [17]}.
